Driving Hamiltonian in a Quantum Searching Problem

We examine the driving Hamiltonian in the analog analogue of Grover’s algorithm by Farhi and Gutmann. For a quantum system with a given oracle Hamiltonian E|w〉〈w|, we explicitly show that while the driving Hamiltonian E|s〉〈s| optimally produces the state |w〉 from an initial state |s〉, the driving Hamiltonian E ′|s〉〈s|(E′ 6= E) does not provide any speedup compared even with a classical computation. PACS number(s):03.67.Lx ∗E-mail: [email protected]